Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efﬁcient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive reﬁnement. We propose a ﬂexible ﬁnite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral ﬁnite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their ﬂexible computation and efﬁcient evaluation. The versatility of our approach
is demonstrated on cutting and adaptive reﬁnement within a simulation framework for corotated linear elasticity.